# American Institute of Mathematical Sciences

November  2019, 39(11): 6485-6506. doi: 10.3934/dcds.2019281

## On the Gevrey regularity of solutions to the 3D ideal MHD equations

 1 Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, 430062 Wuhan, China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, 211106 Nanjing, China 3 Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France

* Corresponding author: Feng Cheng

Received  December 2018 Revised  May 2019 Published  August 2019

In this paper, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevrey radius for the solution of MHD equation.

Citation: Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
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##### References:
 [1] Fucai Li, Zhipeng Zhang. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4279-4304. doi: 10.3934/dcds.2018187 [2] Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101 [3] Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [4] Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305 [5] Mei-Qin Zhan. Gevrey class regularity for the solutions of the Phase-Lock equations of Superconductivity. Conference Publications, 2001, 2001 (Special) : 406-415. doi: 10.3934/proc.2001.2001.406 [6] Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507 [7] Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018 [8] Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763 [9] Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634 [10] Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455 [11] Yanmin Mu. Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2014, 7 (4) : 739-753. doi: 10.3934/krm.2014.7.739 [12] Hua Chen, Wei-Xi Li, Chao-Jiang Xu. Propagation of Gevrey regularity for solutions of Landau equations. Kinetic & Related Models, 2008, 1 (3) : 355-368. doi: 10.3934/krm.2008.1.355 [13] Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309 [14] Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813 [15] Fei Jiang, Song Jiang, Weiwei Wang. Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1853-1898. doi: 10.3934/dcdss.2016076 [16] Eduard Feireisl, Antonin Novotny, Yongzhong Sun. Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 121-143. doi: 10.3934/dcds.2014.34.121 [17] Stuart S. Antman. Regularity properties of planar motions of incompressible rods. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 481-494. doi: 10.3934/dcdsb.2003.3.481 [18] Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873 [19] Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307 [20] Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359

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